Constrained low rank matrix recovery: from efficient algorithms to brain network imaging

The proposed research concerns the development of efficient computational algorithms to factorise matrix data into a low-rank representation, where the factors satisfy several constraints. Two scenarios are of interest:

1) The entire data-matrix is known and the goal is the decomposition of the data into explanatory components that reveal underlying data structure.
2) The data is only partially observed and the decomposition is also used to recover the un-observed data.

1) is often used to remove noise from data (e.g. to clean up images ) or to decompose data into several distinct components (e.g. to separation different speakers in a recording), while 2) is used, for example, by online retailers, who use recommender system to recommend products based on previous purchases or in medical imaging, where we want to reduce a patient's exposure to radiation. In general, better matrix factorisation techniques will enable us to a) acquire data faster, safer and cheaper; b) acquire data at a higher resolution; and c) find better interpretations of data in terms of meaningful underlying factors.

These improvements will be made possible through the development and exploitation of better data models. In particular, we will develop models and algorithms that are able to utilize a range of non-convex constraints, such as sparseness, smoothness, contiguity, block structure and low-rank. Each of these constraints has been individually exploited previously and each was found to be able to capture distinct data features. For example, the usefulness of sparse data models for data recovery has attracted significant attention (e.g. in medical imaging), whilst for matrix data, low-rank models are now becoming widely used (e.g. in recommender systems). We here build on our previous work on the efficient recovery and factorization of data and develop algorithms that can exploit more than one of these constraints. Instead of imposing either sparsity or low-rank, we will develop methods that will enable us to efficiently exploit several constraints jointly. This will have a transformative impact on many applications where data structure can be captured using several constraints, but where each single constraint is not strong enough to offer substantial benefits.

For example, in radio astronomy, observations might be missing, either due to inability to monitor certain regions of the sky or due to inability to physically store the vast amount of data generated by modern radio observatories. The structure in this data is only partially captured by any one constraint and can thus not be fully recovered with current approaches.

Here we are particularly inspired by our current work in functional brain imaging. Magnetic Resonance Imaging (MRI) techniques can be used to measure human brain activity whilst a person is at rest. This type of data provides crucial insights into information processing mechanisms in the living human brain and can also be used to reveal neural mechanisms underlying many brain disorders. Matrix factorization methods are already used as one of the main tools to analyse these data-sets. Current methods construct a low-rank approximation of the spatio-temporal data matrix, describing spatial regions that exhibit joint neural activity, thus revealing several distinct networks of connected brain regions.

Our new methods will significantly improve on current approaches. Advanced data models will allow us to better estimate functional neuro-anatomy and will provide better recovery of under-sampled fMRI data using far fewer measurements. This will speed up data acquisition, reduce cost and provide data of higher quality. This in turn will enable us to develop better techniques to study the healthy human brain as well as to detect and study neural processes that underlie different brain diseases.

Our research is expected to enable and facilitate a wide range of new technologies, many of which are likely to have major future impact in a range of spheres beyond academia. In our digital age where increasing quantities of data are gathered, stored and distributed, whether medical images taken at a hospital or information on purchases made at an online retailer, our ability to acquire, and analyse data is of fundamental importance. Yet, much of the information we have remains incomplete. One canonical example is the recommender system operated by an online bookstore. A database stores the names of titles bought. To recommend books to customers and increase sales, the store would like to estimate which other books a customer might also buy. This problem is essentially that of estimating entries in a large matrix where each row is a customer and each column a book. However, most entries are missing, as each customer normally only buys relatively few books. This is only one of the many problems that can be solved by matrix completion and that can have a profound impact on both the society and the economy.

As with most methodological research, much of the impact of our work will be achieved through secondary channels. To facilitate this process, we will adopt an approach in which traditional academic dissemination will be complemented with a range of direct academic collaborations in more applied fields. To demonstrate how these collaborations can lead to direct impact on society and economy, we present here an impact case study that concentrates on a problem in brain imaging, which has motivated the proposed work and where we have a well-established on-going collaboration with the Oxford Centre for Functional MRI of the Brain.

Society: Impact here will be mainly through improvement of health care and quality of life. MRI techniques, such as those that will directly benefit from our research, have enormous potential as diagnostic tools for the early detection of neurological disorders such as Alzheimer's, autism, schizophrenia and epilepsy. Early detection of these disorders would in turn have ramifications for early treatment. There is therefore clear potential for this research to have a fundamental impact on the significant socio-economic burden imposed by these diseases. Health care provision and subsequent improvement in patient's quality of life will also benefit from pharmacological advances that will be enabled through better techniques to image human brain processes under medical intervention.

Economy: Any improvements in diagnosis and treatment of neurological diseases can lead to substantial financial savings in healthcare provision, especially with our aging population, where mental conditions such as dementia are becoming a significant problem. Additional economic impact can come from the commercialisation of the techniques themselves. Not only is the manufacturing of superconducting magnets used in MRI scanners a major UK industry, there are also now several UK companies specialising in the analysis of drug trial data, which increasingly includes MRI data. Economic impact will be facilitated through close collaboration with the Oxford Centre for Functional MRI of the Brain, which has long standing research agreements with scanner vendors (Siemens) and the pharmaceutical industry (e.g. Pfizer).

Knowledge: In addition to a better understanding of key mathematical concepts that allow a range of non-convex structures to be exploited in signals and images, there will also be an impact on biology and medicine. MRI techniques are important tools in the study of the living human nervous system, and our improvements are likely to lead to better understanding of many brain processes, both in health and in disease. This impact on knowledge will ultimately further our understanding and treatment of disorders such as autism, schizophrenia and epilepsy, which are thought to be diseases of connectivity.